Sufficient conditions for strong structural controllability of uncertain linear time-varying systems

In this paper, we extend the notion of strong structural controllability of linear time-invariant systems, a property that requires the controllability of each system in a specific class given by the zero-nonzero pattern of the system matrices, to the linear time-varying case ẋ (t) = A(t) · x(t) + B(t) · u(t), where A and B are matrices of analytic functions. It is demonstrated that the requirements for strong structural controllability of linear time-invariant systems are not sufficient for strong structural controllability of linear time-varying systems. Moreover, in the main result of this paper, sufficient conditions for strong structural controllability of linear time-varying systems are given and an algorithm for verifying this property is provided. Since time-invariant systems are included in the class of time-varying system, these conditions are also new sufficient conditions for strong structural controllability of time-invariant systems. Finally, the results are illustrated by means of an example.

[1]  Ferdinand Svaricek,et al.  Strong structural controllability of linear systems revisited , 2011, IEEE Conference on Decision and Control and European Control Conference.

[2]  K. Reinschke,et al.  On strong structural controllability of linear systems , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[3]  Gunther Reissig.  On the maximum rank of Toeplitz block matrices of blocks of a given pattern , 2013, ArXiv.

[4]  Takashi Amemiya,et al.  Controllability and Observability of Linear Time-Invariant Uncertain Systems Irrespective of Bounds of Uncertain Parameters , 2011, IEEE Transactions on Automatic Control.

[5]  L. Silverman,et al.  Controllability and Observability in Time-Variable Linear Systems , 1967 .

[6]  Henry D'Angelo,et al.  Linear time-varying systems : analysis and synthesis , 1970 .

[7]  W. J. Terrell,et al.  Duality, observability, and controllability for linear time-varying descriptor systems , 1991 .

[8]  H. Mayeda,et al.  Strong Structural Controllability , 1979 .

[9]  Ute Feldmann,et al.  A simple and general method for detecting structural inconsistencies in large electrical networks , 2003 .

[10]  U. Feldmann,et al.  Computing the generic index of the circuit equations of linear active networks , 1996, 1996 IEEE International Symposium on Circuits and Systems. Circuits and Systems Connecting the World. ISCAS 96.

[11]  Gunther Reissig,et al.  Characterization of strong structural controllability of uncertain linear time-varying discrete-time systems , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[12]  Ching-tai Lin.  Structural controllability , 1974 .

[13]  Takao Fujii,et al.  On the connection between controllability and stabilizability of linear systems with structural uncertain parameters , 1993, Autom..

[14]  Kurt Johannes Reinschke,et al.  Multivariable Control a Graph-theoretic Approach , 1988 .

[15]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .